Persistent Homology via Finite Topological Spaces

• URL: http://arxiv.org/abs/2512.23348v1
• Date: Mon, 29 Dec 2025 10:14:56 GMT
• Title: Persistent Homology via Finite Topological Spaces
• Abstract summary: We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets.<n>We show that standard poset-level simplifications preserve persistent invariants and prove stability of the resulting persistence diagrams under perturbations of the input metric in a density-based instantiation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are continuous identity maps. By passing functorially to posets and to simplicial complexes via crosscut constructions, we obtain persistence modules without requiring inclusion relations between the resulting complexes. We show that standard poset-level simplifications preserve persistent invariants and prove stability of the resulting persistence diagrams under perturbations of the input metric in a density-based instantiation.

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